LMFDB - Embedded newform 3969.2.a.q.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3969,2,Mod(1,3969)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3969.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3969, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3969 = 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3969.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3,0,3,-3,0,0,6,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$31.6926245622$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	3969.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.879385 q^{2} -1.22668 q^{4} -1.34730 q^{5} +2.83750 q^{8} +1.18479 q^{10} +1.65270 q^{11} +3.36959 q^{13} -0.0418891 q^{16} -0.467911 q^{17} +3.22668 q^{19} +1.65270 q^{20} -1.45336 q^{22} +8.94356 q^{23} -3.18479 q^{25} -2.96316 q^{26} +6.26857 q^{29} -9.23442 q^{31} -5.63816 q^{32} +0.411474 q^{34} +9.23442 q^{37} -2.83750 q^{38} -3.82295 q^{40} -3.41147 q^{41} -4.41147 q^{43} -2.02734 q^{44} -7.86484 q^{46} -9.35504 q^{47} +2.80066 q^{50} -4.13341 q^{52} -0.573978 q^{53} -2.22668 q^{55} -5.51249 q^{58} +10.3969 q^{59} -7.63816 q^{61} +8.12061 q^{62} +5.04189 q^{64} -4.53983 q^{65} +0.596267 q^{67} +0.573978 q^{68} -0.554378 q^{71} -2.04963 q^{73} -8.12061 q^{74} -3.95811 q^{76} -2.40373 q^{79} +0.0564370 q^{80} +3.00000 q^{82} +15.0496 q^{83} +0.630415 q^{85} +3.87939 q^{86} +4.68954 q^{88} -9.08647 q^{89} -10.9709 q^{92} +8.22668 q^{94} -4.34730 q^{95} +1.89899 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 6 q^{8} + 6 q^{11} + 3 q^{13} + 3 q^{16} - 6 q^{17} + 3 q^{19} + 6 q^{20} + 9 q^{22} + 12 q^{23} - 6 q^{25} + 3 q^{26} + 9 q^{29} + 3 q^{31} - 9 q^{34} - 3 q^{37} - 6 q^{38}+ \cdots + 3 q^{97}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 + 6 * q^8 + 6 * q^11 + 3 * q^13 + 3 * q^16 - 6 * q^17 + 3 * q^19 + 6 * q^20 + 9 * q^22 + 12 * q^23 - 6 * q^25 + 3 * q^26 + 9 * q^29 + 3 * q^31 - 9 * q^34 - 3 * q^37 - 6 * q^38 + 9 * q^40 - 3 * q^43 + 15 * q^44 - 3 * q^47 - 6 * q^50 + 21 * q^52 + 6 * q^53 - 9 * q^58 + 3 * q^59 - 6 * q^61 + 30 * q^62 + 12 * q^64 + 15 * q^65 - 12 * q^67 - 6 * q^68 + 9 * q^71 + 21 * q^73 - 30 * q^74 - 15 * q^76 - 21 * q^79 + 15 * q^80 + 9 * q^82 + 18 * q^83 + 9 * q^85 + 6 * q^86 + 27 * q^88 - 12 * q^89 + 3 * q^92 + 18 * q^94 - 12 * q^95 + 3 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.879385	−0.621819	−0.310910	−	0.950439i	$-0.600634\pi$
$2$	−0.879385	−0.621819	−0.310910	+	0.950439i	$0.600634\pi$
$3$	0	0
$3$	0	0
$4$	−1.22668	−0.613341
$5$	−1.34730	−0.602529	−0.301265	−	0.953541i	$-0.597409\pi$
$5$	−1.34730	−0.602529	−0.301265	+	0.953541i	$0.597409\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.83750	1.00321
$9$	0	0
$10$	1.18479	0.374664
$11$	1.65270	0.498309	0.249154	−	0.968464i	$-0.419847\pi$
$11$	1.65270	0.498309	0.249154	+	0.968464i	$0.419847\pi$
$12$	0	0
$13$	3.36959	0.934555	0.467277	−	0.884111i	$-0.345235\pi$
$13$	3.36959	0.934555	0.467277	+	0.884111i	$0.345235\pi$
$14$	0	0
$15$	0	0
$16$	−0.0418891	−0.0104723
$17$	−0.467911	−0.113485	−0.0567426	−	0.998389i	$-0.518071\pi$
$17$	−0.467911	−0.113485	−0.0567426	+	0.998389i	$0.518071\pi$
$18$	0	0
$19$	3.22668	0.740252	0.370126	−	0.928982i	$-0.379315\pi$
$19$	3.22668	0.740252	0.370126	+	0.928982i	$0.379315\pi$
$20$	1.65270	0.369556
$21$	0	0
$22$	−1.45336	−0.309858
$23$	8.94356	1.86486	0.932431	−	0.361348i	$-0.117683\pi$
$23$	8.94356	1.86486	0.932431	+	0.361348i	$0.117683\pi$
$24$	0	0
$25$	−3.18479	−0.636959
$26$	−2.96316	−0.581124
$27$	0	0
$28$	0	0
$29$	6.26857	1.16404	0.582022	−	0.813173i	$-0.302262\pi$
$29$	6.26857	1.16404	0.582022	+	0.813173i	$0.302262\pi$
$30$	0	0
$31$	−9.23442	−1.65855	−0.829276	−	0.558840i	$-0.811247\pi$
$31$	−9.23442	−1.65855	−0.829276	+	0.558840i	$0.811247\pi$
$32$	−5.63816	−0.996695
$33$	0	0
$34$	0.411474	0.0705672
$35$	0	0
$36$	0	0
$37$	9.23442	1.51813	0.759065	−	0.651015i	$-0.225657\pi$
$37$	9.23442	1.51813	0.759065	+	0.651015i	$0.225657\pi$
$38$	−2.83750	−0.460303
$39$	0	0
$40$	−3.82295	−0.604461
$41$	−3.41147	−0.532783	−0.266391	−	0.963865i	$-0.585831\pi$
$41$	−3.41147	−0.532783	−0.266391	+	0.963865i	$0.585831\pi$
$42$	0	0
$43$	−4.41147	−0.672743	−0.336372	−	0.941729i	$-0.609200\pi$
$43$	−4.41147	−0.672743	−0.336372	+	0.941729i	$0.609200\pi$
$44$	−2.02734	−0.305633
$45$	0	0
$46$	−7.86484	−1.15961
$47$	−9.35504	−1.36457	−0.682286	−	0.731085i	$-0.739014\pi$
$47$	−9.35504	−1.36457	−0.682286	+	0.731085i	$0.739014\pi$
$48$	0	0
$49$	0	0
$50$	2.80066	0.396073
$51$	0	0
$52$	−4.13341	−0.573201
$53$	−0.573978	−0.0788419	−0.0394210	−	0.999223i	$-0.512551\pi$
$53$	−0.573978	−0.0788419	−0.0394210	+	0.999223i	$0.512551\pi$
$54$	0	0
$55$	−2.22668	−0.300246
$56$	0	0
$57$	0	0
$58$	−5.51249	−0.723825
$59$	10.3969	1.35356	0.676782	−	0.736183i	$-0.263374\pi$
$59$	10.3969	1.35356	0.676782	+	0.736183i	$0.263374\pi$
$60$	0	0
$61$	−7.63816	−0.977966	−0.488983	−	0.872293i	$-0.662632\pi$
$61$	−7.63816	−0.977966	−0.488983	+	0.872293i	$0.662632\pi$
$62$	8.12061	1.03132
$63$	0	0
$64$	5.04189	0.630236
$65$	−4.53983	−0.563097
$66$	0	0
$67$	0.596267	0.0728456	0.0364228	−	0.999336i	$-0.488404\pi$
$67$	0.596267	0.0728456	0.0364228	+	0.999336i	$0.488404\pi$
$68$	0.573978	0.0696051
$69$	0	0
$70$	0	0
$71$	−0.554378	−0.0657925	−0.0328963	−	0.999459i	$-0.510473\pi$
$71$	−0.554378	−0.0657925	−0.0328963	+	0.999459i	$0.510473\pi$
$72$	0	0
$73$	−2.04963	−0.239891	−0.119946	−	0.992780i	$-0.538272\pi$
$73$	−2.04963	−0.239891	−0.119946	+	0.992780i	$0.538272\pi$
$74$	−8.12061	−0.944002
$75$	0	0
$76$	−3.95811	−0.454026
$77$	0	0
$78$	0	0
$79$	−2.40373	−0.270441	−0.135221	−	0.990816i	$-0.543174\pi$
$79$	−2.40373	−0.270441	−0.135221	+	0.990816i	$0.543174\pi$
$80$	0.0564370	0.00630985
$81$	0	0
$82$	3.00000	0.331295
$83$	15.0496	1.65191	0.825956	−	0.563735i	$-0.190636\pi$
$83$	15.0496	1.65191	0.825956	+	0.563735i	$0.190636\pi$
$84$	0	0
$85$	0.630415	0.0683781
$86$	3.87939	0.418325
$87$	0	0
$88$	4.68954	0.499907
$89$	−9.08647	−0.963164	−0.481582	−	0.876401i	$-0.659938\pi$
$89$	−9.08647	−0.963164	−0.481582	+	0.876401i	$0.659938\pi$
$90$	0	0
$91$	0	0
$92$	−10.9709	−1.14380
$93$	0	0
$94$	8.22668	0.848517
$95$	−4.34730	−0.446023
$96$	0	0
$97$	1.89899	0.192813	0.0964064	−	0.995342i	$-0.469265\pi$
$97$	1.89899	0.192813	0.0964064	+	0.995342i	$0.469265\pi$
$98$	0	0
$99$	0	0
$100$	3.90673	0.390673
$101$	1.70914	0.170066	0.0850329	−	0.996378i	$-0.472900\pi$
$101$	1.70914	0.170066	0.0850329	+	0.996378i	$0.472900\pi$
$102$	0	0
$103$	3.63816	0.358478	0.179239	−	0.983806i	$-0.442636\pi$
$103$	3.63816	0.358478	0.179239	+	0.983806i	$0.442636\pi$
$104$	9.56118	0.937551
$105$	0	0
$106$	0.504748	0.0490254
$107$	7.12836	0.689124	0.344562	−	0.938764i	$-0.388027\pi$
$107$	7.12836	0.689124	0.344562	+	0.938764i	$0.388027\pi$
$108$	0	0
$109$	0.403733	0.0386706	0.0193353	−	0.999813i	$-0.493845\pi$
$109$	0.403733	0.0386706	0.0193353	+	0.999813i	$0.493845\pi$
$110$	1.95811	0.186699
$111$	0	0
$112$	0	0
$113$	14.3696	1.35178	0.675888	−	0.737004i	$-0.263760\pi$
$113$	14.3696	1.35178	0.675888	+	0.737004i	$0.263760\pi$
$114$	0	0
$115$	−12.0496	−1.12363
$116$	−7.68954	−0.713956
$117$	0	0
$118$	−9.14290	−0.841672
$119$	0	0
$120$	0	0
$121$	−8.26857	−0.751688
$122$	6.71688	0.608118
$123$	0	0
$124$	11.3277	1.01726
$125$	11.0273	0.986315
$126$	0	0
$127$	−20.7716	−1.84318	−0.921589	−	0.388167i	$-0.873108\pi$
$127$	−20.7716	−1.84318	−0.921589	+	0.388167i	$0.873108\pi$
$128$	6.84255	0.604802
$129$	0	0
$130$	3.99226	0.350144
$131$	7.16519	0.626026	0.313013	−	0.949749i	$-0.398662\pi$
$131$	7.16519	0.626026	0.313013	+	0.949749i	$0.398662\pi$
$132$	0	0
$133$	0	0
$134$	−0.524348	−0.0452968
$135$	0	0
$136$	−1.32770	−0.113849
$137$	2.56893	0.219478	0.109739	−	0.993960i	$-0.464998\pi$
$137$	2.56893	0.219478	0.109739	+	0.993960i	$0.464998\pi$
$138$	0	0
$139$	6.13341	0.520229	0.260114	−	0.965578i	$-0.416240\pi$
$139$	6.13341	0.520229	0.260114	+	0.965578i	$0.416240\pi$
$140$	0	0
$141$	0	0
$142$	0.487511	0.0409111
$143$	5.56893	0.465697
$144$	0	0
$145$	−8.44562	−0.701371
$146$	1.80241	0.149169
$147$	0	0
$148$	−11.3277	−0.931131
$149$	0.431074	0.0353150	0.0176575	−	0.999844i	$-0.494379\pi$
$149$	0.431074	0.0353150	0.0176575	+	0.999844i	$0.494379\pi$
$150$	0	0
$151$	−2.47060	−0.201055	−0.100527	−	0.994934i	$-0.532053\pi$
$151$	−2.47060	−0.201055	−0.100527	+	0.994934i	$0.532053\pi$
$152$	9.15570	0.742625
$153$	0	0
$154$	0	0
$155$	12.4415	0.999326
$156$	0	0
$157$	−10.1334	−0.808734	−0.404367	−	0.914597i	$-0.632508\pi$
$157$	−10.1334	−0.808734	−0.404367	+	0.914597i	$0.632508\pi$
$158$	2.11381	0.168166
$159$	0	0
$160$	7.59627	0.600538
$161$	0	0
$162$	0	0
$163$	−2.59627	−0.203355	−0.101678	−	0.994817i	$-0.532421\pi$
$163$	−2.59627	−0.203355	−0.101678	+	0.994817i	$0.532421\pi$
$164$	4.18479	0.326777
$165$	0	0
$166$	−13.2344	−1.02719
$167$	23.1830	1.79396	0.896979	−	0.442074i	$-0.145757\pi$
$167$	23.1830	1.79396	0.896979	+	0.442074i	$0.145757\pi$
$168$	0	0
$169$	−1.64590	−0.126607
$170$	−0.554378	−0.0425188
$171$	0	0
$172$	5.41147	0.412621
$173$	4.75196	0.361285	0.180643	−	0.983549i	$-0.442182\pi$
$173$	4.75196	0.361285	0.180643	+	0.983549i	$0.442182\pi$
$174$	0	0
$175$	0	0
$176$	−0.0692302	−0.00521842
$177$	0	0
$178$	7.99050	0.598914
$179$	−8.53209	−0.637718	−0.318859	−	0.947802i	$-0.603300\pi$
$179$	−8.53209	−0.637718	−0.318859	+	0.947802i	$0.603300\pi$
$180$	0	0
$181$	17.2344	1.28102	0.640512	−	0.767948i	$-0.278722\pi$
$181$	17.2344	1.28102	0.640512	+	0.767948i	$0.278722\pi$
$182$	0	0
$183$	0	0
$184$	25.3773	1.87084
$185$	−12.4415	−0.914718
$186$	0	0
$187$	−0.773318	−0.0565506
$188$	11.4757	0.836948
$189$	0	0
$190$	3.82295	0.277346
$191$	12.9094	0.934092	0.467046	−	0.884233i	$-0.345318\pi$
$191$	12.9094	0.934092	0.467046	+	0.884233i	$0.345318\pi$
$192$	0	0
$193$	−0.638156	−0.0459355	−0.0229677	−	0.999736i	$-0.507311\pi$
$193$	−0.638156	−0.0459355	−0.0229677	+	0.999736i	$0.507311\pi$
$194$	−1.66994	−0.119895
$195$	0	0
$196$	0	0
$197$	11.4456	0.815467	0.407733	−	0.913101i	$-0.366319\pi$
$197$	11.4456	0.815467	0.407733	+	0.913101i	$0.366319\pi$
$198$	0	0
$199$	3.63816	0.257902	0.128951	−	0.991651i	$-0.458839\pi$
$199$	3.63816	0.257902	0.128951	+	0.991651i	$0.458839\pi$
$200$	−9.03684	−0.639001
$201$	0	0
$202$	−1.50299	−0.105750
$203$	0	0
$204$	0	0
$205$	4.59627	0.321017
$206$	−3.19934	−0.222909
$207$	0	0
$208$	−0.141149	−0.00978691
$209$	5.33275	0.368874
$210$	0	0
$211$	5.82295	0.400868	0.200434	−	0.979707i	$-0.435765\pi$
$211$	5.82295	0.400868	0.200434	+	0.979707i	$0.435765\pi$
$212$	0.704088	0.0483570
$213$	0	0
$214$	−6.26857	−0.428511
$215$	5.94356	0.405348
$216$	0	0
$217$	0	0
$218$	−0.355037	−0.0240461
$219$	0	0
$220$	2.73143	0.184153
$221$	−1.57667	−0.106058
$222$	0	0
$223$	−7.08378	−0.474365	−0.237182	−	0.971465i	$-0.576224\pi$
$223$	−7.08378	−0.474365	−0.237182	+	0.971465i	$0.576224\pi$
$224$	0	0
$225$	0	0
$226$	−12.6364	−0.840561
$227$	11.9436	0.792722	0.396361	−	0.918095i	$-0.370273\pi$
$227$	11.9436	0.792722	0.396361	+	0.918095i	$0.370273\pi$
$228$	0	0
$229$	17.5526	1.15991	0.579955	−	0.814649i	$-0.303070\pi$
$229$	17.5526	1.15991	0.579955	+	0.814649i	$0.303070\pi$
$230$	10.5963	0.698697
$231$	0	0
$232$	17.7870	1.16778
$233$	16.2540	1.06484	0.532418	−	0.846481i	$-0.321283\pi$
$233$	16.2540	1.06484	0.532418	+	0.846481i	$0.321283\pi$
$234$	0	0
$235$	12.6040	0.822195
$236$	−12.7537	−0.830196
$237$	0	0
$238$	0	0
$239$	−15.0993	−0.976690	−0.488345	−	0.872651i	$-0.662399\pi$
$239$	−15.0993	−0.976690	−0.488345	+	0.872651i	$0.662399\pi$
$240$	0	0
$241$	15.6382	1.00734	0.503671	−	0.863896i	$-0.331982\pi$
$241$	15.6382	1.00734	0.503671	+	0.863896i	$0.331982\pi$
$242$	7.27126	0.467414
$243$	0	0
$244$	9.36959	0.599826
$245$	0	0
$246$	0	0
$247$	10.8726	0.691806
$248$	−26.2026	−1.66387
$249$	0	0
$250$	−9.69728	−0.613310
$251$	−19.0651	−1.20338	−0.601690	−	0.798730i	$-0.705506\pi$
$251$	−19.0651	−1.20338	−0.601690	+	0.798730i	$0.705506\pi$
$252$	0	0
$253$	14.7811	0.929277
$254$	18.2662	1.14612
$255$	0	0
$256$	−16.1010	−1.00631
$257$	−26.5817	−1.65812	−0.829061	−	0.559158i	$-0.811124\pi$
$257$	−26.5817	−1.65812	−0.829061	+	0.559158i	$0.811124\pi$
$258$	0	0
$259$	0	0
$260$	5.56893	0.345370
$261$	0	0
$262$	−6.30096	−0.389275
$263$	−0.734118	−0.0452676	−0.0226338	−	0.999744i	$-0.507205\pi$
$263$	−0.734118	−0.0452676	−0.0226338	+	0.999744i	$0.507205\pi$
$264$	0	0
$265$	0.773318	0.0475046
$266$	0	0
$267$	0	0
$268$	−0.731429	−0.0446792
$269$	20.8503	1.27126	0.635632	−	0.771992i	$-0.280739\pi$
$269$	20.8503	1.27126	0.635632	+	0.771992i	$0.280739\pi$
$270$	0	0
$271$	−6.95811	−0.422675	−0.211338	−	0.977413i	$-0.567782\pi$
$271$	−6.95811	−0.422675	−0.211338	+	0.977413i	$0.567782\pi$
$272$	0.0196004	0.00118845
$273$	0	0
$274$	−2.25908	−0.136476
$275$	−5.26352	−0.317402
$276$	0	0
$277$	17.8726	1.07386	0.536930	−	0.843627i	$-0.319584\pi$
$277$	17.8726	1.07386	0.536930	+	0.843627i	$0.319584\pi$
$278$	−5.39363	−0.323488
$279$	0	0
$280$	0	0
$281$	22.3105	1.33093	0.665465	−	0.746429i	$-0.268233\pi$
$281$	22.3105	1.33093	0.665465	+	0.746429i	$0.268233\pi$
$282$	0	0
$283$	18.5945	1.10533	0.552665	−	0.833404i	$-0.313611\pi$
$283$	18.5945	1.10533	0.552665	+	0.833404i	$0.313611\pi$
$284$	0.680045	0.0403532
$285$	0	0
$286$	−4.89723	−0.289579
$287$	0	0
$288$	0	0
$289$	−16.7811	−0.987121
$290$	7.42696	0.436126
$291$	0	0
$292$	2.51424	0.147135
$293$	−13.0915	−0.764815	−0.382407	−	0.923994i	$-0.624905\pi$
$293$	−13.0915	−0.764815	−0.382407	+	0.923994i	$0.624905\pi$
$294$	0	0
$295$	−14.0077	−0.815562
$296$	26.2026	1.52300
$297$	0	0
$298$	−0.379081	−0.0219595
$299$	30.1361	1.74282
$300$	0	0
$301$	0	0
$302$	2.17261	0.125020
$303$	0	0
$304$	−0.135163	−0.00775211
$305$	10.2909	0.589253
$306$	0	0
$307$	−6.31046	−0.360157	−0.180078	−	0.983652i	$-0.557635\pi$
$307$	−6.31046	−0.360157	−0.180078	+	0.983652i	$0.557635\pi$
$308$	0	0
$309$	0	0
$310$	−10.9409	−0.621400
$311$	9.52435	0.540076	0.270038	−	0.962850i	$-0.412964\pi$
$311$	9.52435	0.540076	0.270038	+	0.962850i	$0.412964\pi$
$312$	0	0
$313$	17.6287	0.996431	0.498215	−	0.867053i	$-0.333989\pi$
$313$	17.6287	0.996431	0.498215	+	0.867053i	$0.333989\pi$
$314$	8.91117	0.502886
$315$	0	0
$316$	2.94862	0.165873
$317$	8.07697	0.453648	0.226824	−	0.973936i	$-0.427166\pi$
$317$	8.07697	0.453648	0.226824	+	0.973936i	$0.427166\pi$
$318$	0	0
$319$	10.3601	0.580054
$320$	−6.79292	−0.379736
$321$	0	0
$322$	0	0
$323$	−1.50980	−0.0840075
$324$	0	0
$325$	−10.7314	−0.595273
$326$	2.28312	0.126450
$327$	0	0
$328$	−9.68004	−0.534491
$329$	0	0
$330$	0	0
$331$	23.0496	1.26692	0.633461	−	0.773775i	$-0.281634\pi$
$331$	23.0496	1.26692	0.633461	+	0.773775i	$0.281634\pi$
$332$	−18.4611	−1.01318
$333$	0	0
$334$	−20.3868	−1.11552
$335$	−0.803348	−0.0438916
$336$	0	0
$337$	29.0232	1.58100	0.790498	−	0.612465i	$-0.209822\pi$
$337$	29.0232	1.58100	0.790498	+	0.612465i	$0.209822\pi$
$338$	1.44738	0.0787269
$339$	0	0
$340$	−0.773318	−0.0419391
$341$	−15.2618	−0.826471
$342$	0	0
$343$	0	0
$344$	−12.5175	−0.674901
$345$	0	0
$346$	−4.17881	−0.224654
$347$	12.9463	0.694991	0.347496	−	0.937682i	$-0.387032\pi$
$347$	12.9463	0.694991	0.347496	+	0.937682i	$0.387032\pi$
$348$	0	0
$349$	−1.46286	−0.0783050	−0.0391525	−	0.999233i	$-0.512466\pi$
$349$	−1.46286	−0.0783050	−0.0391525	+	0.999233i	$0.512466\pi$
$350$	0	0
$351$	0	0
$352$	−9.31820	−0.496662
$353$	−14.3327	−0.762855	−0.381428	−	0.924399i	$-0.624567\pi$
$353$	−14.3327	−0.762855	−0.381428	+	0.924399i	$0.624567\pi$
$354$	0	0
$355$	0.746911	0.0396419
$356$	11.1462	0.590747
$357$	0	0
$358$	7.50299	0.396546
$359$	−20.9368	−1.10500	−0.552500	−	0.833513i	$-0.686326\pi$
$359$	−20.9368	−1.10500	−0.552500	+	0.833513i	$0.686326\pi$
$360$	0	0
$361$	−8.58853	−0.452028
$362$	−15.1557	−0.796566
$363$	0	0
$364$	0	0
$365$	2.76146	0.144541
$366$	0	0
$367$	12.0574	0.629390	0.314695	−	0.949193i	$-0.398098\pi$
$367$	12.0574	0.629390	0.314695	+	0.949193i	$0.398098\pi$
$368$	−0.374638	−0.0195293
$369$	0	0
$370$	10.9409	0.568789
$371$	0	0
$372$	0	0
$373$	−0.781059	−0.0404417	−0.0202209	−	0.999796i	$-0.506437\pi$
$373$	−0.781059	−0.0404417	−0.0202209	+	0.999796i	$0.506437\pi$
$374$	0.680045	0.0351643
$375$	0	0
$376$	−26.5449	−1.36895
$377$	21.1225	1.08786
$378$	0	0
$379$	−6.92396	−0.355660	−0.177830	−	0.984061i	$-0.556908\pi$
$379$	−6.92396	−0.355660	−0.177830	+	0.984061i	$0.556908\pi$
$380$	5.33275	0.273564
$381$	0	0
$382$	−11.3523	−0.580837
$383$	−7.73236	−0.395105	−0.197553	−	0.980292i	$-0.563299\pi$
$383$	−7.73236	−0.395105	−0.197553	+	0.980292i	$0.563299\pi$
$384$	0	0
$385$	0	0
$386$	0.561185	0.0285636
$387$	0	0
$388$	−2.32945	−0.118260
$389$	5.39961	0.273771	0.136886	−	0.990587i	$-0.456291\pi$
$389$	5.39961	0.273771	0.136886	+	0.990587i	$0.456291\pi$
$390$	0	0
$391$	−4.18479	−0.211634
$392$	0	0
$393$	0	0
$394$	−10.0651	−0.507073
$395$	3.23854	0.162949
$396$	0	0
$397$	29.2344	1.46723	0.733617	−	0.679563i	$-0.237831\pi$
$397$	29.2344	1.46723	0.733617	+	0.679563i	$0.237831\pi$
$398$	−3.19934	−0.160368
$399$	0	0
$400$	0.133408	0.00667040
$401$	−27.3979	−1.36818	−0.684092	−	0.729396i	$-0.739801\pi$
$401$	−27.3979	−1.36818	−0.684092	+	0.729396i	$0.739801\pi$
$402$	0	0
$403$	−31.1162	−1.55001
$404$	−2.09657	−0.104308
$405$	0	0
$406$	0	0
$407$	15.2618	0.756498
$408$	0	0
$409$	9.02498	0.446256	0.223128	−	0.974789i	$-0.428373\pi$
$409$	9.02498	0.446256	0.223128	+	0.974789i	$0.428373\pi$
$410$	−4.04189	−0.199615
$411$	0	0
$412$	−4.46286	−0.219869
$413$	0	0
$414$	0	0
$415$	−20.2763	−0.995325
$416$	−18.9982	−0.931466
$417$	0	0
$418$	−4.68954	−0.229373
$419$	−0.175297	−0.00856382	−0.00428191	−	0.999991i	$-0.501363\pi$
$419$	−0.175297	−0.00856382	−0.00428191	+	0.999991i	$0.501363\pi$
$420$	0	0
$421$	−24.7050	−1.20405	−0.602025	−	0.798478i	$-0.705639\pi$
$421$	−24.7050	−1.20405	−0.602025	+	0.798478i	$0.705639\pi$
$422$	−5.12061	−0.249268
$423$	0	0
$424$	−1.62866	−0.0790947
$425$	1.49020	0.0722853
$426$	0	0
$427$	0	0
$428$	−8.74422	−0.422668
$429$	0	0
$430$	−5.22668	−0.252053
$431$	−29.3191	−1.41225	−0.706126	−	0.708086i	$-0.749559\pi$
$431$	−29.3191	−1.41225	−0.706126	+	0.708086i	$0.749559\pi$
$432$	0	0
$433$	−19.6554	−0.944578	−0.472289	−	0.881444i	$-0.656572\pi$
$433$	−19.6554	−0.944578	−0.472289	+	0.881444i	$0.656572\pi$
$434$	0	0
$435$	0	0
$436$	−0.495252	−0.0237183
$437$	28.8580	1.38047
$438$	0	0
$439$	21.9299	1.04666	0.523330	−	0.852130i	$-0.324690\pi$
$439$	21.9299	1.04666	0.523330	+	0.852130i	$0.324690\pi$
$440$	−6.31820	−0.301208
$441$	0	0
$442$	1.38650	0.0659489
$443$	−18.7101	−0.888942	−0.444471	−	0.895793i	$-0.646608\pi$
$443$	−18.7101	−0.888942	−0.444471	+	0.895793i	$0.646608\pi$
$444$	0	0
$445$	12.2422	0.580334
$446$	6.22937	0.294969
$447$	0	0
$448$	0	0
$449$	6.68004	0.315251	0.157625	−	0.987499i	$-0.449616\pi$
$449$	6.68004	0.315251	0.157625	+	0.987499i	$0.449616\pi$
$450$	0	0
$451$	−5.63816	−0.265490
$452$	−17.6269	−0.829100
$453$	0	0
$454$	−10.5030	−0.492930
$455$	0	0
$456$	0	0
$457$	−19.4287	−0.908837	−0.454418	−	0.890788i	$-0.650153\pi$
$457$	−19.4287	−0.908837	−0.454418	+	0.890788i	$0.650153\pi$
$458$	−15.4355	−0.721254
$459$	0	0
$460$	14.7811	0.689170
$461$	0.965852	0.0449842	0.0224921	−	0.999747i	$-0.492840\pi$
$461$	0.965852	0.0449842	0.0224921	+	0.999747i	$0.492840\pi$
$462$	0	0
$463$	−0.445622	−0.0207098	−0.0103549	−	0.999946i	$-0.503296\pi$
$463$	−0.445622	−0.0207098	−0.0103549	+	0.999946i	$0.503296\pi$
$464$	−0.262585	−0.0121902
$465$	0	0
$466$	−14.2935	−0.662136
$467$	34.2148	1.58327	0.791637	−	0.610992i	$-0.209229\pi$
$467$	34.2148	1.58327	0.791637	+	0.610992i	$0.209229\pi$
$468$	0	0
$469$	0	0
$470$	−11.0838	−0.511257
$471$	0	0
$472$	29.5012	1.35790
$473$	−7.29086	−0.335234
$474$	0	0
$475$	−10.2763	−0.471510
$476$	0	0
$477$	0	0
$478$	13.2781	0.607325
$479$	21.7929	0.995744	0.497872	−	0.867251i	$-0.334115\pi$
$479$	21.7929	0.995744	0.497872	+	0.867251i	$0.334115\pi$
$480$	0	0
$481$	31.1162	1.41878
$482$	−13.7520	−0.626385
$483$	0	0
$484$	10.1429	0.461041
$485$	−2.55850	−0.116175
$486$	0	0
$487$	19.3928	0.878772	0.439386	−	0.898298i	$-0.355196\pi$
$487$	19.3928	0.878772	0.439386	+	0.898298i	$0.355196\pi$
$488$	−21.6732	−0.981101
$489$	0	0
$490$	0	0
$491$	26.1566	1.18043	0.590216	−	0.807245i	$-0.299043\pi$
$491$	26.1566	1.18043	0.590216	+	0.807245i	$0.299043\pi$
$492$	0	0
$493$	−2.93313	−0.132102
$494$	−9.56118	−0.430178
$495$	0	0
$496$	0.386821	0.0173688
$497$	0	0
$498$	0	0
$499$	−14.3013	−0.640214	−0.320107	−	0.947381i	$-0.603719\pi$
$499$	−14.3013	−0.640214	−0.320107	+	0.947381i	$0.603719\pi$
$500$	−13.5270	−0.604947
$501$	0	0
$502$	16.7656	0.748284
$503$	−18.7033	−0.833937	−0.416969	−	0.908921i	$-0.636908\pi$
$503$	−18.7033	−0.833937	−0.416969	+	0.908921i	$0.636908\pi$
$504$	0	0
$505$	−2.30272	−0.102470
$506$	−12.9982	−0.577842
$507$	0	0
$508$	25.4801	1.13050
$509$	25.6091	1.13510	0.567551	−	0.823338i	$-0.307891\pi$
$509$	25.6091	1.13510	0.567551	+	0.823338i	$0.307891\pi$
$510$	0	0
$511$	0	0
$512$	0.473897	0.0209435
$513$	0	0
$514$	23.3756	1.03105
$515$	−4.90167	−0.215994
$516$	0	0
$517$	−15.4611	−0.679979
$518$	0	0
$519$	0	0
$520$	−12.8817	−0.564902
$521$	21.2121	0.929320	0.464660	−	0.885489i	$-0.346176\pi$
$521$	21.2121	0.929320	0.464660	+	0.885489i	$0.346176\pi$
$522$	0	0
$523$	−20.8057	−0.909770	−0.454885	−	0.890550i	$-0.650320\pi$
$523$	−20.8057	−0.909770	−0.454885	+	0.890550i	$0.650320\pi$
$524$	−8.78941	−0.383967
$525$	0	0
$526$	0.645572	0.0281483
$527$	4.32089	0.188221
$528$	0	0
$529$	56.9873	2.47771
$530$	−0.680045	−0.0295393
$531$	0	0
$532$	0	0
$533$	−11.4953	−0.497915
$534$	0	0
$535$	−9.60401	−0.415217
$536$	1.69190	0.0730791
$537$	0	0
$538$	−18.3354	−0.790497
$539$	0	0
$540$	0	0
$541$	26.7297	1.14920	0.574599	−	0.818435i	$-0.305158\pi$
$541$	26.7297	1.14920	0.574599	+	0.818435i	$0.305158\pi$
$542$	6.11886	0.262828
$543$	0	0
$544$	2.63816	0.113110
$545$	−0.543948	−0.0233002
$546$	0	0
$547$	36.7624	1.57185	0.785923	−	0.618324i	$-0.212188\pi$
$547$	36.7624	1.57185	0.785923	+	0.618324i	$0.212188\pi$
$548$	−3.15125	−0.134615
$549$	0	0
$550$	4.62866	0.197367
$551$	20.2267	0.861686
$552$	0	0
$553$	0	0
$554$	−15.7169	−0.667746
$555$	0	0
$556$	−7.52374	−0.319078
$557$	32.3387	1.37024	0.685118	−	0.728432i	$-0.259751\pi$
$557$	32.3387	1.37024	0.685118	+	0.728432i	$0.259751\pi$
$558$	0	0
$559$	−14.8648	−0.628716
$560$	0	0
$561$	0	0
$562$	−19.6195	−0.827598
$563$	17.7419	0.747730	0.373865	−	0.927483i	$-0.378032\pi$
$563$	17.7419	0.747730	0.373865	+	0.927483i	$0.378032\pi$
$564$	0	0
$565$	−19.3601	−0.814485
$566$	−16.3517	−0.687315
$567$	0	0
$568$	−1.57304	−0.0660035
$569$	−26.6013	−1.11519	−0.557593	−	0.830115i	$-0.688275\pi$
$569$	−26.6013	−1.11519	−0.557593	+	0.830115i	$0.688275\pi$
$570$	0	0
$571$	−10.0172	−0.419208	−0.209604	−	0.977786i	$-0.567218\pi$
$571$	−10.0172	−0.419208	−0.209604	+	0.977786i	$0.567218\pi$
$572$	−6.83130	−0.285631
$573$	0	0
$574$	0	0
$575$	−28.4834	−1.18784
$576$	0	0
$577$	32.9145	1.37025	0.685124	−	0.728427i	$-0.259748\pi$
$577$	32.9145	1.37025	0.685124	+	0.728427i	$0.259748\pi$
$578$	14.7570	0.613811
$579$	0	0
$580$	10.3601	0.430179
$581$	0	0
$582$	0	0
$583$	−0.948615	−0.0392876
$584$	−5.81582	−0.240660
$585$	0	0
$586$	11.5125	0.475577
$587$	15.0729	0.622123	0.311062	−	0.950390i	$-0.399315\pi$
$587$	15.0729	0.622123	0.311062	+	0.950390i	$0.399315\pi$
$588$	0	0
$589$	−29.7965	−1.22775
$590$	12.3182	0.507132
$591$	0	0
$592$	−0.386821	−0.0158983
$593$	−41.0009	−1.68371	−0.841853	−	0.539706i	$-0.818535\pi$
$593$	−41.0009	−1.68371	−0.841853	+	0.539706i	$0.818535\pi$
$594$	0	0
$595$	0	0
$596$	−0.528791	−0.0216601
$597$	0	0
$598$	−26.5012	−1.08372
$599$	6.07367	0.248164	0.124082	−	0.992272i	$-0.460401\pi$
$599$	6.07367	0.248164	0.124082	+	0.992272i	$0.460401\pi$
$600$	0	0
$601$	14.1352	0.576585	0.288293	−	0.957542i	$-0.406912\pi$
$601$	14.1352	0.576585	0.288293	+	0.957542i	$0.406912\pi$
$602$	0	0
$603$	0	0
$604$	3.03064	0.123315
$605$	11.1402	0.452914
$606$	0	0
$607$	−46.0898	−1.87073	−0.935363	−	0.353689i	$-0.884927\pi$
$607$	−46.0898	−1.87073	−0.935363	+	0.353689i	$0.884927\pi$
$608$	−18.1925	−0.737805
$609$	0	0
$610$	−9.04963	−0.366409
$611$	−31.5226	−1.27527
$612$	0	0
$613$	−26.4938	−1.07008	−0.535038	−	0.844828i	$-0.679703\pi$
$613$	−26.4938	−1.07008	−0.535038	+	0.844828i	$0.679703\pi$
$614$	5.54933	0.223953
$615$	0	0
$616$	0	0
$617$	−2.24990	−0.0905777	−0.0452889	−	0.998974i	$-0.514421\pi$
$617$	−2.24990	−0.0905777	−0.0452889	+	0.998974i	$0.514421\pi$
$618$	0	0
$619$	−6.19078	−0.248828	−0.124414	−	0.992230i	$-0.539705\pi$
$619$	−6.19078	−0.248828	−0.124414	+	0.992230i	$0.539705\pi$
$620$	−15.2618	−0.612927
$621$	0	0
$622$	−8.37557	−0.335830
$623$	0	0
$624$	0	0
$625$	1.06687	0.0426746
$626$	−15.5024	−0.619600
$627$	0	0
$628$	12.4305	0.496030
$629$	−4.32089	−0.172285
$630$	0	0
$631$	26.1661	1.04166	0.520829	−	0.853661i	$-0.325623\pi$
$631$	26.1661	1.04166	0.520829	+	0.853661i	$0.325623\pi$
$632$	−6.82058	−0.271308
$633$	0	0
$634$	−7.10277	−0.282087
$635$	27.9855	1.11057
$636$	0	0
$637$	0	0
$638$	−9.11051	−0.360689
$639$	0	0
$640$	−9.21894	−0.364411
$641$	4.88888	0.193099	0.0965496	−	0.995328i	$-0.469219\pi$
$641$	4.88888	0.193099	0.0965496	+	0.995328i	$0.469219\pi$
$642$	0	0
$643$	40.3678	1.59195	0.795976	−	0.605328i	$-0.206958\pi$
$643$	40.3678	1.59195	0.795976	+	0.605328i	$0.206958\pi$
$644$	0	0
$645$	0	0
$646$	1.32770	0.0522375
$647$	2.28075	0.0896657	0.0448329	−	0.998995i	$-0.485724\pi$
$647$	2.28075	0.0896657	0.0448329	+	0.998995i	$0.485724\pi$
$648$	0	0
$649$	17.1830	0.674493
$650$	9.43706	0.370152
$651$	0	0
$652$	3.18479	0.124726
$653$	23.4793	0.918815	0.459407	−	0.888226i	$-0.348062\pi$
$653$	23.4793	0.918815	0.459407	+	0.888226i	$0.348062\pi$
$654$	0	0
$655$	−9.65364	−0.377199
$656$	0.142903	0.00557944
$657$	0	0
$658$	0	0
$659$	−47.9623	−1.86835	−0.934174	−	0.356818i	$-0.883862\pi$
$659$	−47.9623	−1.86835	−0.934174	+	0.356818i	$0.883862\pi$
$660$	0	0
$661$	−29.3090	−1.13999	−0.569995	−	0.821648i	$-0.693055\pi$
$661$	−29.3090	−1.13999	−0.569995	+	0.821648i	$0.693055\pi$
$662$	−20.2695	−0.787797
$663$	0	0
$664$	42.7033	1.65721
$665$	0	0
$666$	0	0
$667$	56.0634	2.17078
$668$	−28.4382	−1.10031
$669$	0	0
$670$	0.706452	0.0272926
$671$	−12.6236	−0.487329
$672$	0	0
$673$	26.3182	1.01449	0.507246	−	0.861801i	$-0.330664\pi$
$673$	26.3182	1.01449	0.507246	+	0.861801i	$0.330664\pi$
$674$	−25.5226	−0.983094
$675$	0	0
$676$	2.01899	0.0776535
$677$	−35.8907	−1.37939	−0.689697	−	0.724098i	$-0.742256\pi$
$677$	−35.8907	−1.37939	−0.689697	+	0.724098i	$0.742256\pi$
$678$	0	0
$679$	0	0
$680$	1.78880	0.0685973
$681$	0	0
$682$	13.4210	0.513915
$683$	35.0642	1.34169	0.670847	−	0.741596i	$-0.265931\pi$
$683$	35.0642	1.34169	0.670847	+	0.741596i	$0.265931\pi$
$684$	0	0
$685$	−3.46110	−0.132242
$686$	0	0
$687$	0	0
$688$	0.184793	0.00704515
$689$	−1.93407	−0.0736821
$690$	0	0
$691$	−2.06687	−0.0786273	−0.0393136	−	0.999227i	$-0.512517\pi$
$691$	−2.06687	−0.0786273	−0.0393136	+	0.999227i	$0.512517\pi$
$692$	−5.82915	−0.221591
$693$	0	0
$694$	−11.3847	−0.432159
$695$	−8.26352	−0.313453
$696$	0	0
$697$	1.59627	0.0604629
$698$	1.28642	0.0486916
$699$	0	0
$700$	0	0
$701$	−7.36009	−0.277987	−0.138993	−	0.990293i	$-0.544387\pi$
$701$	−7.36009	−0.277987	−0.138993	+	0.990293i	$0.544387\pi$
$702$	0	0
$703$	29.7965	1.12380
$704$	8.33275	0.314052
$705$	0	0
$706$	12.6040	0.474358
$707$	0	0
$708$	0	0
$709$	9.10876	0.342086	0.171043	−	0.985264i	$-0.445286\pi$
$709$	9.10876	0.342086	0.171043	+	0.985264i	$0.445286\pi$
$710$	−0.656822	−0.0246501
$711$	0	0
$712$	−25.7828	−0.966252
$713$	−82.5886	−3.09297
$714$	0	0
$715$	−7.50299	−0.280596
$716$	10.4662	0.391139
$717$	0	0
$718$	18.4115	0.687110
$719$	25.9537	0.967908	0.483954	−	0.875093i	$-0.339200\pi$
$719$	25.9537	0.967908	0.483954	+	0.875093i	$0.339200\pi$
$720$	0	0
$721$	0	0
$722$	7.55262	0.281080
$723$	0	0
$724$	−21.1411	−0.785705
$725$	−19.9641	−0.741448
$726$	0	0
$727$	10.1601	0.376819	0.188409	−	0.982091i	$-0.439667\pi$
$727$	10.1601	0.376819	0.188409	+	0.982091i	$0.439667\pi$
$728$	0	0
$729$	0	0
$730$	−2.42839	−0.0898786
$731$	2.06418	0.0763464
$732$	0	0
$733$	−40.6614	−1.50186	−0.750931	−	0.660381i	$-0.770395\pi$
$733$	−40.6614	−1.50186	−0.750931	+	0.660381i	$0.770395\pi$
$734$	−10.6031	−0.391367
$735$	0	0
$736$	−50.4252	−1.85870
$737$	0.985452	0.0362996
$738$	0	0
$739$	−25.3618	−0.932951	−0.466475	−	0.884534i	$-0.654476\pi$
$739$	−25.3618	−0.932951	−0.466475	+	0.884534i	$0.654476\pi$
$740$	15.2618	0.561034
$741$	0	0
$742$	0	0
$743$	−22.4442	−0.823398	−0.411699	−	0.911320i	$-0.635064\pi$
$743$	−22.4442	−0.823398	−0.411699	+	0.911320i	$0.635064\pi$
$744$	0	0
$745$	−0.580785	−0.0212783
$746$	0.686852	0.0251474
$747$	0	0
$748$	0.948615	0.0346848
$749$	0	0
$750$	0	0
$751$	24.2172	0.883698	0.441849	−	0.897090i	$-0.354323\pi$
$751$	24.2172	0.883698	0.441849	+	0.897090i	$0.354323\pi$
$752$	0.391874	0.0142902
$753$	0	0
$754$	−18.5748	−0.676454
$755$	3.32863	0.121141
$756$	0	0
$757$	9.11793	0.331397	0.165698	−	0.986176i	$-0.447012\pi$
$757$	9.11793	0.331397	0.165698	+	0.986176i	$0.447012\pi$
$758$	6.08883	0.221156
$759$	0	0
$760$	−12.3354	−0.447453
$761$	18.2722	0.662366	0.331183	−	0.943566i	$-0.392552\pi$
$761$	18.2722	0.662366	0.331183	+	0.943566i	$0.392552\pi$
$762$	0	0
$763$	0	0
$764$	−15.8357	−0.572917
$765$	0	0
$766$	6.79973	0.245684
$767$	35.0333	1.26498
$768$	0	0
$769$	−18.5294	−0.668187	−0.334094	−	0.942540i	$-0.608430\pi$
$769$	−18.5294	−0.668187	−0.334094	+	0.942540i	$0.608430\pi$
$770$	0	0
$771$	0	0
$772$	0.782814	0.0281741
$773$	2.96080	0.106493	0.0532463	−	0.998581i	$-0.483043\pi$
$773$	2.96080	0.106493	0.0532463	+	0.998581i	$0.483043\pi$
$774$	0	0
$775$	29.4097	1.05643
$776$	5.38836	0.193431
$777$	0	0
$778$	−4.74834	−0.170236
$779$	−11.0077	−0.394393
$780$	0	0
$781$	−0.916222	−0.0327850
$782$	3.68004	0.131598
$783$	0	0
$784$	0	0
$785$	13.6527	0.487286
$786$	0	0
$787$	33.4020	1.19065	0.595326	−	0.803484i	$-0.297023\pi$
$787$	33.4020	1.19065	0.595326	+	0.803484i	$0.297023\pi$
$788$	−14.0401	−0.500159
$789$	0	0
$790$	−2.84793	−0.101325
$791$	0	0
$792$	0	0
$793$	−25.7374	−0.913962
$794$	−25.7083	−0.912354
$795$	0	0
$796$	−4.46286	−0.158182
$797$	−49.3509	−1.74810	−0.874050	−	0.485837i	$-0.838515\pi$
$797$	−49.3509	−1.74810	−0.874050	+	0.485837i	$0.838515\pi$
$798$	0	0
$799$	4.37733	0.154859
$800$	17.9564	0.634853
$801$	0	0
$802$	24.0933	0.850763
$803$	−3.38743	−0.119540
$804$	0	0
$805$	0	0
$806$	27.3631	0.963824
$807$	0	0
$808$	4.84968	0.170611
$809$	19.8280	0.697115	0.348558	−	0.937287i	$-0.386672\pi$
$809$	19.8280	0.697115	0.348558	+	0.937287i	$0.386672\pi$
$810$	0	0
$811$	23.8557	0.837686	0.418843	−	0.908059i	$-0.362436\pi$
$811$	23.8557	0.837686	0.418843	+	0.908059i	$0.362436\pi$
$812$	0	0
$813$	0	0
$814$	−13.4210	−0.470405
$815$	3.49794	0.122528
$816$	0	0
$817$	−14.2344	−0.497999
$818$	−7.93643	−0.277491
$819$	0	0
$820$	−5.63816	−0.196893
$821$	−50.9427	−1.77791	−0.888957	−	0.457991i	$-0.848569\pi$
$821$	−50.9427	−1.77791	−0.888957	+	0.457991i	$0.848569\pi$
$822$	0	0
$823$	13.6149	0.474587	0.237293	−	0.971438i	$-0.423740\pi$
$823$	13.6149	0.474587	0.237293	+	0.971438i	$0.423740\pi$
$824$	10.3233	0.359628
$825$	0	0
$826$	0	0
$827$	36.2158	1.25935	0.629673	−	0.776861i	$-0.283189\pi$
$827$	36.2158	1.25935	0.629673	+	0.776861i	$0.283189\pi$
$828$	0	0
$829$	−25.3259	−0.879606	−0.439803	−	0.898094i	$-0.644952\pi$
$829$	−25.3259	−0.879606	−0.439803	+	0.898094i	$0.644952\pi$
$830$	17.8307	0.618912
$831$	0	0
$832$	16.9891	0.588990
$833$	0	0
$834$	0	0
$835$	−31.2344	−1.08091
$836$	−6.54158	−0.226245
$837$	0	0
$838$	0.154154	0.00532515
$839$	−8.71419	−0.300847	−0.150424	−	0.988622i	$-0.548064\pi$
$839$	−8.71419	−0.300847	−0.150424	+	0.988622i	$0.548064\pi$
$840$	0	0
$841$	10.2950	0.354999
$842$	21.7252	0.748701
$843$	0	0
$844$	−7.14290	−0.245869
$845$	2.21751	0.0762847
$846$	0	0
$847$	0	0
$848$	0.0240434	0.000825654 0
$849$	0	0
$850$	−1.31046	−0.0449484
$851$	82.5886	2.83110
$852$	0	0
$853$	11.9813	0.410233	0.205117	−	0.978738i	$-0.434243\pi$
$853$	11.9813	0.410233	0.205117	+	0.978738i	$0.434243\pi$
$854$	0	0
$855$	0	0
$856$	20.2267	0.691334
$857$	−6.50030	−0.222046	−0.111023	−	0.993818i	$-0.535413\pi$
$857$	−6.50030	−0.222046	−0.111023	+	0.993818i	$0.535413\pi$
$858$	0	0
$859$	53.5526	1.82719	0.913596	−	0.406623i	$-0.133294\pi$
$859$	53.5526	1.82719	0.913596	+	0.406623i	$0.133294\pi$
$860$	−7.29086	−0.248616
$861$	0	0
$862$	25.7828	0.878166
$863$	3.69965	0.125937	0.0629687	−	0.998016i	$-0.479943\pi$
$863$	3.69965	0.125937	0.0629687	+	0.998016i	$0.479943\pi$
$864$	0	0
$865$	−6.40230	−0.217685
$866$	17.2847	0.587357
$867$	0	0
$868$	0	0
$869$	−3.97266	−0.134763
$870$	0	0
$871$	2.00917	0.0680782
$872$	1.14559	0.0387946
$873$	0	0
$874$	−25.3773	−0.858401
$875$	0	0
$876$	0	0
$877$	−11.7888	−0.398079	−0.199040	−	0.979991i	$-0.563782\pi$
$877$	−11.7888	−0.398079	−0.199040	+	0.979991i	$0.563782\pi$
$878$	−19.2849	−0.650833
$879$	0	0
$880$	0.0932736	0.00314425
$881$	−49.4858	−1.66722	−0.833609	−	0.552355i	$-0.813729\pi$
$881$	−49.4858	−1.66722	−0.833609	+	0.552355i	$0.813729\pi$
$882$	0	0
$883$	−21.5357	−0.724734	−0.362367	−	0.932035i	$-0.618031\pi$
$883$	−21.5357	−0.724734	−0.362367	+	0.932035i	$0.618031\pi$
$884$	1.93407	0.0650497
$885$	0	0
$886$	16.4534	0.552762
$887$	−11.8848	−0.399051	−0.199526	−	0.979893i	$-0.563940\pi$
$887$	−11.8848	−0.399051	−0.199526	+	0.979893i	$0.563940\pi$
$888$	0	0
$889$	0	0
$890$	−10.7656	−0.360863
$891$	0	0
$892$	8.68954	0.290947
$893$	−30.1857	−1.01013
$894$	0	0
$895$	11.4953	0.384244
$896$	0	0
$897$	0	0
$898$	−5.87433	−0.196029
$899$	−57.8866	−1.93063
$900$	0	0
$901$	0.268571	0.00894739
$902$	4.95811	0.165087
$903$	0	0
$904$	40.7736	1.35611
$905$	−23.2199	−0.771855
$906$	0	0
$907$	−26.0215	−0.864029	−0.432014	−	0.901867i	$-0.642197\pi$
$907$	−26.0215	−0.864029	−0.432014	+	0.901867i	$0.642197\pi$
$908$	−14.6509	−0.486209
$909$	0	0
$910$	0	0
$911$	−4.03272	−0.133610	−0.0668050	−	0.997766i	$-0.521281\pi$
$911$	−4.03272	−0.133610	−0.0668050	+	0.997766i	$0.521281\pi$
$912$	0	0
$913$	24.8726	0.823162
$914$	17.0853	0.565132
$915$	0	0
$916$	−21.5315	−0.711420
$917$	0	0
$918$	0	0
$919$	27.4270	0.904732	0.452366	−	0.891832i	$-0.350580\pi$
$919$	27.4270	0.904732	0.452366	+	0.891832i	$0.350580\pi$
$920$	−34.1908	−1.12724
$921$	0	0
$922$	−0.849356	−0.0279720
$923$	−1.86802	−0.0614867
$924$	0	0
$925$	−29.4097	−0.966986
$926$	0.391874	0.0128778
$927$	0	0
$928$	−35.3432	−1.16020
$929$	−7.67675	−0.251866	−0.125933	−	0.992039i	$-0.540192\pi$
$929$	−7.67675	−0.251866	−0.125933	+	0.992039i	$0.540192\pi$
$930$	0	0
$931$	0	0
$932$	−19.9385	−0.653108
$933$	0	0
$934$	−30.0880	−0.984510
$935$	1.04189	0.0340734
$936$	0	0
$937$	2.02465	0.0661425	0.0330713	−	0.999453i	$-0.489471\pi$
$937$	2.02465	0.0661425	0.0330713	+	0.999453i	$0.489471\pi$
$938$	0	0
$939$	0	0
$940$	−15.4611	−0.504286
$941$	6.13928	0.200135	0.100067	−	0.994981i	$-0.468094\pi$
$941$	6.13928	0.200135	0.100067	+	0.994981i	$0.468094\pi$
$942$	0	0
$943$	−30.5107	−0.993566
$944$	−0.435518	−0.0141749
$945$	0	0
$946$	6.41147	0.208455
$947$	5.56448	0.180821	0.0904107	−	0.995905i	$-0.471182\pi$
$947$	5.56448	0.180821	0.0904107	+	0.995905i	$0.471182\pi$
$948$	0	0
$949$	−6.90640	−0.224191
$950$	9.03684	0.293194
$951$	0	0
$952$	0	0
$953$	−8.72018	−0.282474	−0.141237	−	0.989976i	$-0.545108\pi$
$953$	−8.72018	−0.282474	−0.141237	+	0.989976i	$0.545108\pi$
$954$	0	0
$955$	−17.3928	−0.562818
$956$	18.5220	0.599044
$957$	0	0
$958$	−19.1644	−0.619173
$959$	0	0
$960$	0	0
$961$	54.2746	1.75079
$962$	−27.3631	−0.882222
$963$	0	0
$964$	−19.1830	−0.617844
$965$	0.859785	0.0276775
$966$	0	0
$967$	−57.7698	−1.85775	−0.928876	−	0.370391i	$-0.879224\pi$
$967$	−57.7698	−1.85775	−0.928876	+	0.370391i	$0.879224\pi$
$968$	−23.4620	−0.754098
$969$	0	0
$970$	2.24990	0.0722401
$971$	30.7192	0.985828	0.492914	−	0.870078i	$-0.335932\pi$
$971$	30.7192	0.985828	0.492914	+	0.870078i	$0.335932\pi$
$972$	0	0
$973$	0	0
$974$	−17.0537	−0.546437
$975$	0	0
$976$	0.319955	0.0102415
$977$	10.3000	0.329527	0.164764	−	0.986333i	$-0.447314\pi$
$977$	10.3000	0.329527	0.164764	+	0.986333i	$0.447314\pi$
$978$	0	0
$979$	−15.0172	−0.479953
$980$	0	0
$981$	0	0
$982$	−23.0018	−0.734015
$983$	−13.6963	−0.436846	−0.218423	−	0.975854i	$-0.570091\pi$
$983$	−13.6963	−0.436846	−0.218423	+	0.975854i	$0.570091\pi$
$984$	0	0
$985$	−15.4206	−0.491343
$986$	2.57935	0.0821434
$987$	0	0
$988$	−13.3372	−0.424313
$989$	−39.4543	−1.25457
$990$	0	0
$991$	57.9813	1.84184	0.920919	−	0.389754i	$-0.127440\pi$
$991$	57.9813	1.84184	0.920919	+	0.389754i	$0.127440\pi$
$992$	52.0651	1.65307
$993$	0	0
$994$	0	0
$995$	−4.90167	−0.155394
$996$	0	0
$997$	−16.2175	−0.513614	−0.256807	−	0.966463i	$-0.582670\pi$
$997$	−16.2175	−0.513614	−0.256807	+	0.966463i	$0.582670\pi$
$998$	12.5763	0.398097
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.q.1.1		3
3.2	odd	2		3969.2.a.l.1.3		3
7.6	odd	2		567.2.a.h.1.1		3
9.2	odd	6		1323.2.f.d.442.1		6
9.4	even	3		441.2.f.c.295.3		6
9.5	odd	6		1323.2.f.d.883.1		6
9.7	even	3		441.2.f.c.148.3		6
21.20	even	2		567.2.a.c.1.3		3
28.27	even	2		9072.2.a.ca.1.2		3
63.2	odd	6		1323.2.g.e.361.1		6
63.4	even	3		441.2.g.b.79.3		6
63.5	even	6		1323.2.h.c.802.3		6
63.11	odd	6		1323.2.h.b.226.3		6
63.13	odd	6		63.2.f.a.43.3	yes	6
63.16	even	3		441.2.g.b.67.3		6
63.20	even	6		189.2.f.b.64.1		6
63.23	odd	6		1323.2.h.b.802.3		6
63.25	even	3		441.2.h.e.373.1		6
63.31	odd	6		441.2.g.c.79.3		6
63.32	odd	6		1323.2.g.e.667.1		6
63.34	odd	6		63.2.f.a.22.3	✓	6
63.38	even	6		1323.2.h.c.226.3		6
63.40	odd	6		441.2.h.d.214.1		6
63.41	even	6		189.2.f.b.127.1		6
63.47	even	6		1323.2.g.d.361.1		6
63.52	odd	6		441.2.h.d.373.1		6
63.58	even	3		441.2.h.e.214.1		6
63.59	even	6		1323.2.g.d.667.1		6
63.61	odd	6		441.2.g.c.67.3		6
84.83	odd	2		9072.2.a.bs.1.2		3
252.83	odd	6		3024.2.r.k.1009.2		6
252.139	even	6		1008.2.r.h.673.2		6
252.167	odd	6		3024.2.r.k.2017.2		6
252.223	even	6		1008.2.r.h.337.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.f.a.22.3	✓	6	63.34	odd	6
63.2.f.a.43.3	yes	6	63.13	odd	6
189.2.f.b.64.1		6	63.20	even	6
189.2.f.b.127.1		6	63.41	even	6
441.2.f.c.148.3		6	9.7	even	3
441.2.f.c.295.3		6	9.4	even	3
441.2.g.b.67.3		6	63.16	even	3
441.2.g.b.79.3		6	63.4	even	3
441.2.g.c.67.3		6	63.61	odd	6
441.2.g.c.79.3		6	63.31	odd	6
441.2.h.d.214.1		6	63.40	odd	6
441.2.h.d.373.1		6	63.52	odd	6
441.2.h.e.214.1		6	63.58	even	3
441.2.h.e.373.1		6	63.25	even	3
567.2.a.c.1.3		3	21.20	even	2
567.2.a.h.1.1		3	7.6	odd	2
1008.2.r.h.337.2		6	252.223	even	6
1008.2.r.h.673.2		6	252.139	even	6
1323.2.f.d.442.1		6	9.2	odd	6
1323.2.f.d.883.1		6	9.5	odd	6
1323.2.g.d.361.1		6	63.47	even	6
1323.2.g.d.667.1		6	63.59	even	6
1323.2.g.e.361.1		6	63.2	odd	6
1323.2.g.e.667.1		6	63.32	odd	6
1323.2.h.b.226.3		6	63.11	odd	6
1323.2.h.b.802.3		6	63.23	odd	6
1323.2.h.c.226.3		6	63.38	even	6
1323.2.h.c.802.3		6	63.5	even	6
3024.2.r.k.1009.2		6	252.83	odd	6
3024.2.r.k.2017.2		6	252.167	odd	6
3969.2.a.l.1.3		3	3.2	odd	2
3969.2.a.q.1.1		3	1.1	even	1	trivial
9072.2.a.bs.1.2		3	84.83	odd	2
9072.2.a.ca.1.2		3	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

\(f(q)\)	\(=\)	\(q-0.879385 q^{2} -1.22668 q^{4} -1.34730 q^{5} +2.83750 q^{8} +1.18479 q^{10} +1.65270 q^{11} +3.36959 q^{13} -0.0418891 q^{16} -0.467911 q^{17} +3.22668 q^{19} +1.65270 q^{20} -1.45336 q^{22} +8.94356 q^{23} -3.18479 q^{25} -2.96316 q^{26} +6.26857 q^{29} -9.23442 q^{31} -5.63816 q^{32} +0.411474 q^{34} +9.23442 q^{37} -2.83750 q^{38} -3.82295 q^{40} -3.41147 q^{41} -4.41147 q^{43} -2.02734 q^{44} -7.86484 q^{46} -9.35504 q^{47} +2.80066 q^{50} -4.13341 q^{52} -0.573978 q^{53} -2.22668 q^{55} -5.51249 q^{58} +10.3969 q^{59} -7.63816 q^{61} +8.12061 q^{62} +5.04189 q^{64} -4.53983 q^{65} +0.596267 q^{67} +0.573978 q^{68} -0.554378 q^{71} -2.04963 q^{73} -8.12061 q^{74} -3.95811 q^{76} -2.40373 q^{79} +0.0564370 q^{80} +3.00000 q^{82} +15.0496 q^{83} +0.630415 q^{85} +3.87939 q^{86} +4.68954 q^{88} -9.08647 q^{89} -10.9709 q^{92} +8.22668 q^{94} -4.34730 q^{95} +1.89899 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 6 q^{8} + 6 q^{11} + 3 q^{13} + 3 q^{16} - 6 q^{17} + 3 q^{19} + 6 q^{20} + 9 q^{22} + 12 q^{23} - 6 q^{25} + 3 q^{26} + 9 q^{29} + 3 q^{31} - 9 q^{34} - 3 q^{37} - 6 q^{38}+ \cdots + 3 q^{97}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 + 6 * q^8 + 6 * q^11 + 3 * q^13 + 3 * q^16 - 6 * q^17 + 3 * q^19 + 6 * q^20 + 9 * q^22 + 12 * q^23 - 6 * q^25 + 3 * q^26 + 9 * q^29 + 3 * q^31 - 9 * q^34 - 3 * q^37 - 6 * q^38 + 9 * q^40 - 3 * q^43 + 15 * q^44 - 3 * q^47 - 6 * q^50 + 21 * q^52 + 6 * q^53 - 9 * q^58 + 3 * q^59 - 6 * q^61 + 30 * q^62 + 12 * q^64 + 15 * q^65 - 12 * q^67 - 6 * q^68 + 9 * q^71 + 21 * q^73 - 30 * q^74 - 15 * q^76 - 21 * q^79 + 15 * q^80 + 9 * q^82 + 18 * q^83 + 9 * q^85 + 6 * q^86 + 27 * q^88 - 12 * q^89 + 3 * q^92 + 18 * q^94 - 12 * q^95 + 3 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more